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K-stars LDP: A Novel Framework for (p, q)-clique Enumeration under Local Differential Privacy

Henan Sun, Zhengyu Wu, Rong-Hua Li, Guoren Wang, Zening Li

TL;DR

The paper addresses private counting of $(p,q)$-cliques in bipartite graphs under Local Differential Privacy, identifying high noise in edge-LDP approaches as a core limitation. It introduces a $k$-stars LDP framework that obfuscates the existence of $k$-stars rather than individual edges, leveraging two-round user-collector interaction, Warner's Randomized Response, absolute value correction, and $k$-stars sampling to reduce estimation error. The authors prove that $ ext{$ ext{epsilon}$-}k$-stars LDP implies a bounded leakage relative to edge LDP, establish unbiasedness of the estimators, and derive variance bounds demonstrating improved utility, especially for larger $(p,q)$. Empirical results on four public bipartite graphs show that $k$-stars LDP outperforms edge LDP across metrics and settings, with pronounced gains in sparse graphs and when enumerating more complex $(p,q)$-cliques, validating its practical impact for privacy-preserving subgraph analysis.

Abstract

(p,q)-clique enumeration on a bipartite graph is critical for calculating clustering coefficient and detecting densest subgraph. It is necessary to carry out subgraph enumeration while protecting users' privacy from any potential attacker as the count of subgraph may contain sensitive information. Most recent studies focus on the privacy protection algorithms based on edge LDP (Local Differential Privacy). However, these algorithms suffer a large estimation error due to the great amount of required noise. In this paper, we propose a novel idea of k-stars LDP and a novel k-stars LDP algorithm for (p, q)-clique enumeration with a small estimation error, where a k-stars is a star-shaped graph with k nodes connecting to one node. The effectiveness of edge LDP relies on its capacity to obfuscate the existence of an edge between the user and his one-hop neighbors. This is based on the premise that a user should be aware of the existence of his one-hop neighbors. Similarly, we can apply this premise to k-stars as well, where an edge is a specific genre of 1-stars. Based on this fact, we first propose the k-stars neighboring list to enable our algorithm to obfuscate the existence of k-stars with Warner' s RR. Then, we propose the absolute value correction technique and the k-stars sampling technique to further reduce the estimation error. Finally, with the two-round user-collector interaction mechanism, we propose our k-stars LDP algorithm to count the number of (p, q)-clique while successfully protecting users' privacy. Both the theoretical analysis and experiments have showed the superiority of our algorithm over the algorithms based on edge LDP.

K-stars LDP: A Novel Framework for (p, q)-clique Enumeration under Local Differential Privacy

TL;DR

The paper addresses private counting of -cliques in bipartite graphs under Local Differential Privacy, identifying high noise in edge-LDP approaches as a core limitation. It introduces a -stars LDP framework that obfuscates the existence of -stars rather than individual edges, leveraging two-round user-collector interaction, Warner's Randomized Response, absolute value correction, and -stars sampling to reduce estimation error. The authors prove that ext{epsilon}-stars LDP implies a bounded leakage relative to edge LDP, establish unbiasedness of the estimators, and derive variance bounds demonstrating improved utility, especially for larger . Empirical results on four public bipartite graphs show that -stars LDP outperforms edge LDP across metrics and settings, with pronounced gains in sparse graphs and when enumerating more complex -cliques, validating its practical impact for privacy-preserving subgraph analysis.

Abstract

(p,q)-clique enumeration on a bipartite graph is critical for calculating clustering coefficient and detecting densest subgraph. It is necessary to carry out subgraph enumeration while protecting users' privacy from any potential attacker as the count of subgraph may contain sensitive information. Most recent studies focus on the privacy protection algorithms based on edge LDP (Local Differential Privacy). However, these algorithms suffer a large estimation error due to the great amount of required noise. In this paper, we propose a novel idea of k-stars LDP and a novel k-stars LDP algorithm for (p, q)-clique enumeration with a small estimation error, where a k-stars is a star-shaped graph with k nodes connecting to one node. The effectiveness of edge LDP relies on its capacity to obfuscate the existence of an edge between the user and his one-hop neighbors. This is based on the premise that a user should be aware of the existence of his one-hop neighbors. Similarly, we can apply this premise to k-stars as well, where an edge is a specific genre of 1-stars. Based on this fact, we first propose the k-stars neighboring list to enable our algorithm to obfuscate the existence of k-stars with Warner' s RR. Then, we propose the absolute value correction technique and the k-stars sampling technique to further reduce the estimation error. Finally, with the two-round user-collector interaction mechanism, we propose our k-stars LDP algorithm to count the number of (p, q)-clique while successfully protecting users' privacy. Both the theoretical analysis and experiments have showed the superiority of our algorithm over the algorithms based on edge LDP.
Paper Structure (17 sections, 4 theorems, 25 equations, 13 figures, 3 tables, 2 algorithms)

This paper contains 17 sections, 4 theorems, 25 equations, 13 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Notations
  5. Local Differential Privacy on Graphs
  6. Two-round User-collector Interaction Mechanism
  7. Utility Discussion
  8. Locally Differential Privacy Algorithms for $(p,q)$-clique Enumeration
  9. Baseline Algorithm of Edge LDP
  10. K-stars LDP Algorithm
  11. Motivation
  12. Algorithm
  13. Theoretical Analysis
  14. Experiments
  15. Datasets
  16. ...and 2 more sections

Key Result

Theorem 1

If a local randomized algorithm $\mathcal{R}$ provides $\epsilon$-$k$-stars LDP, it also provides $\frac{k2^{(k-1)}}{2^k-1}\epsilon$-edge LDP.

Figures (13)

  • Figure 1: An example of $(p,q)$-clique.
  • Figure 2: The example of $(3,3)$-clique. To count $(3,3)$-clique, traditional edge LDP algorithms require 6 noisy edges (6 random variables), depicted as red edges on the left. However, these noisy edges can be viewed as 2 noisy $3$-stars (2 random variables), depicted as yellow one and brown one on the right.
  • Figure 3: The overview of two-round user-collector interaction mechanism.
  • Figure 4: The examples of $(p,q)$-clique and near $(p,q)$-clique.
  • Figure 5: An example of $k$-stars neighboring list $(k=2)$.
  • ...and 8 more figures

Theorems & Definitions (10)

  • Definition 1: $\epsilon$-edge LDP imola2022communication
  • Definition 2: $k$-stars neighboring list
  • Definition 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • Theorem 3
  • proof